program 60-450—internal gear line of action tooth plot ... · pdf fileinvolute spur and...

Program 60-450—Internal Gear Line of Action Tooth Plot

Introduction The primary purpose of this model is to provide an accurate set of polar and rectangular coordinates for use in plotting the final form of external and internal involute spur and helical gears singly and in mesh. In addition to a plot of the teeth the model furnishes numerical results for many of the design parameters of interest to the designer. The model can be used for the design of a gear set and the necessary tooling or as a “final visual” check of the detailed design processes that were used to produce the design. The pinion may be hobbed with non-topping, semi-topping, tip relief or topping hobs, shaped with pinion type cutters or defined by a basic rack form for molding. The internal gear may be shaped or defined by an involute and a circular arc fillet. If the gears are post processed after hobbing or shaping, provision has been made to accommodate the finishing stock. IT IS ASSUMED THAT NO STEPS WERE CUT IN THE INVOLUTE OR FILLET BY THE FINISH TOOLS. The model will warn you if the finish stock on the side of the tooth is greater than the protuberance but the step produced on the plot of the tooth is not necessarily at the actual location of the step produced by the finish tool. Only if there is enough natural undercut (small pinions) will a satisfactory gear be produced under these conditions. These conditions must be checked by programs such as UTS Program 500 for the pinion. For pinions that are fillet ground with a V-Type grinding wheel the dimensions of the wheel may be substituted for the hob. The hob (or basic rack) flank angle is measured from a normal to the hob (or basic rack) reference line. The shaper cutter tooth thickness at the shaper reference pitch diameter, the protuberance and the tip radius are measured in the normal plane. The model checks for trochoidal interference (tooth tip interference with the internal gear ID) for both the pinion and the internal gear shaper cutter if used. Trim interference (as cutter is fed radially into the gear) is also checked for the internal gear shaper cutter. The model will produce coordinates at intervals of about 0.050 inch divided by the Operating Normal Pitch (near the fillet/involute intersection the fillet coordinates are about one tenth of this). For example, a gear set with an operating normal pitch of one the coordinates would be spaced at about 0.050-inch intervals. For an operating normal pitch of 10 the coordinates would be spaced at about 0.005-inch intervals. The radius to the intersection between the involute flank and the fillet of the tooth will be calculated and marked on the plot of the tooth (if “mark” is set to 'y) in addition to being displayed in the output column on the Variable Sheet.

UTS Integrated Gear Software

2

The coordinates for spur gears are the actual coordinates of the gears produced by the specified tools. For helical gears the coordinates are produced for the virtual spur gears in the operating normal plane. (In some very low tooth number helical pinions the error between the tooth thickness at the effective outside diameter or the involute-fillet tangent point and the virtual spur tooth prevent an accurate plot of the teeth. In this case it is suggested that models 60-411, 60-108, 60-410 and 60-460 be used to analyze the gear set. The model will notify you of this condition.) The model contains some warning messages that may stop execution of the program (“Root Interference”, for example). If it is desired to “visually” check the gears when these messages appear it is only necessary to temporarily cancel the rule or statement producing the message. (The rule or statement producing the message will be marked with > in the status column.) Some messages occurring at the end of model execution will cause TK to indicate that the model is not resolved. This will be the case if the gear set has an undesirable condition or a condition making the set useless. The solution indicator at the right side of the Status Bar, located at the bottom of the screen, will then not read “OK”. You will still get valid data on the Variable Sheet and, usually, a plot of the teeth. Other warning messages may appear that only halt execution to notify you of some condition. The gear data to audit a design should be available from your production data. The hob or shaper data is available from the tool supplier. Because the actual calculations are done in a TK Solver model, the model can be worked directly in the the TK Solver Variable Sheet. In this instance, it is recommended that the input field for “Number of teeth on plot” and “Driver contact roll angle” be left blank until the rest of the data is satisfactory. Then use the Plot Configuration tab of the data entry form, or enter the plot configuration data directly in the TK Solver Variable Sheet, and solve again to obtain a plot of the teeth in mesh.

60-450—Internal Gear Line of Action Tooth Plot

3

Examples

The following examples are meant to aid in learning to use this model. Many design factors, including speed, load and ambient conditions, must be considered to optimize a design. Therefore, these examples are not meant to be optimized designs and should not be used as rigid “patterns” for actual designs.

Example 1

Example 1 is a spur gear set with a “high-addendum” 12 tooth pinion cut on a 13 tooth blank meshed with a “standard” 66 tooth internal gear. The pinion is hobbed with a non-topping finish hob. The gear is cut with a finishing shaper cutter. The pinion is the driver. We will check for undercut and interference at the first and last points of contact. Since the pinion is the driver the first point of contact will be at the pinion root (gear tip) and the last point of contact will be at the pinion tip (gear root).

Open a new analysis in model 60-450. Make certain that the units are “US”. Figure 1A shows the completed data input form; Report 1A gives the inputs and outputs of the solved model.


4

Fig. 1A


5

Report 1A

Model Title : Program 60-450 Unit System: US

PINION, number of teeth 12

Hobbed ('hob), Shaped ('shp), Formed hob ('frm)

INTERNAL GEAR, number of teeth 66

Shaped ('shp) or Formed ('frm) shp

NORMAL PLANE

Diametral Pitch 10.000000 1/in `

Nominal pressure angle 20.000000 deg

Module 2.540000 mm `

Base pitch 0.2952 in

NORMAL PLANE PINION

Finished tooth thickness at Ref PD 0.1915 in

Total circular finish stock on tooth 0.0000 in thickness

NORMAL PLANE INTERNAL GEAR


Finished_space width at Ref PD 0.1599 in


TRANSVERSE PLANE

Diametral_Pitch 10.000000 1/in `


6


Nominal_pressure angle 20.000000 deg


Base_pitch 0.2952 in

Pinion_tooth thickness at Ref PD 0.1915 in

Gear_tooth thickness at Ref PD 0.1543 in

Gear_space width at Ref PD 0.1599 in

COMMON

Helix angle 0.000000 deg

Base_helix angle 0.0000 deg

Axial pitch _ in

Operating_center distance 2.65000 in

Standard_center distance 2.7000 in

Net face width 1.1000 in

PINION

Outside Diameter 1.5000 in

Normal_top land width 0.0260 in

Start_Tip Modification NA in

Roll_at_Start of Tip Modification NA deg

Normal_OD tip relief NA in

Normal_circular OD tip relief NA in

Transverse_circular_OD tip relief NA in

Effective_outside diameter 1.5000 in

Normal_effective OD tip relief NA in

Normal_tooth_thickness_at_Eff OD 0.0260 in

Pointed tooth diameter (No tip mod) 1.5293 in


7


Reference PD 1.2000 in

Inv/fillet intersection dia (TIF) 1.1288 in

Roll_at_inv/fill intersection dia 2.659 deg

Normal_TT_at_inv/fill intersection dia 0.1969 in

Minimum_fillet radius 0.0293 in

Root diameter 1.0446 in

Whole depth of tooth (from Eff OD) 0.2277 in

Base_diameter 1.1276 in

Lead _ in

Area of space (Normal section) 0.04 in^2

Area of tooth (Normal section) 0.03 in^2

INTERNAL GEAR

Inside (minor) Diameter 6.4100 in


Minimum ID w/o involute interference 6.3880 in

Start_Tip_Modification NA in


Normal_ID tip relief NA in

Normal_circular_ID tip relief NA in

Transverse_circular_ID tip relief NA in

Effective_inside diameter 6.4100 in

Normal_effective_ID tip relief NA in

Normal_Tooth Thickness at eff ID 0.0909 in

Pointed_tooth diameter (No tip mod) Below_BD in



8






Major diameter 6.8575 in

Whole depth of tooth (from Eff ID) 0.2238 in


Lead _ in



OPERATING DATA

Change in Operating CD from "Std" CD -0.0500 in

Working depth of active flanks 0.1950 in

Total_working depth 0.1950 in

Transverse_circular backlash 0.0020 in

Normal_diametral pitch 10.188679 1/in `

Transverse_diametral pitch 10.188679 1/in `

Normal_pressure angle 16.780000 deg

Transverse_pressure angle 16.780000 deg

Helix_angle 0.000000 deg

OPERATING DATA PINION

Pitch_diameter 1.1778 in

Normal_tooth thickness 0.1953 in

Transverse_tooth thickness 0.1953 in


9


Angular backlash 0.193 deg

Start of active profile (SAP) 1.1312 in

Root_Clearance 0.0327 in

Max_specific sliding 2.283

OPERATING DATA INTERNAL GEAR



Normal_space width 0.1973 in


Transverse_space width 0.1973 in


Start_of active profile (SAP) 6.6941 in

Root_clearance 0.0288 in


PLOT CONFIGURATION

Mark inv/fil intersections? y

Mark mod/inv intersections? y

Number of teeth on plot 1

Pinion_contact roll angle of deg

Pinion tooth number 1

PINION ROLL ANGLES

Start_of_active profile-No Undercut 4.558 deg

Actual_start of active profile 4.558 deg

Lowest_single tooth contact-Spur 20.260 deg


10


Lowest_double tooth contact-HCRS deg

Operating pitch point 17.276 deg

Highest_single tooth contact-Spur 34.558 deg

Highest_double tooth contact-HCRS deg

Actual_end of active profile 50.260 deg

Effective outside diameter 50.260 deg

GEAR ROLL ANGLES

Effective inside diameter 14.964 deg


Start_of_active_profile 23.273 deg

CONTACT DATA

Total_arc of contact - No Undercut 45.702 deg

Arc_of_approach 12.718 deg

Arc_of_recess 32.984 deg

Approach action 27.83 %

Recess action 72.17 %

Profile contact ratio 1.523

Actual profile contact ratio 1.523

Contact past finished involute? No

Helical contact ratio 0.000

Total_contact ratio 1.523

Unmodified Profile Contact Ratio NA

PINION HOB

Hob type n


11


Flank angle 20.0000 deg

Tip to Reference Line 0.1250 in

Tooth thickness at Reference Line 0.1571 in

Tip_radius 0.0250 in

Protuberance 0.0000 in

Protuberance_angle from flank _ deg

Protuberance_pressure angle _ deg

Tip_to_flank/prot intersection _ in

PINION HOB SEMI-TOPPING AND TIP RELIEF

Reference_Line_to Start Mod Ramp _ in

Pressure Angle of Mod Ramp _ deg

PINION HOB TOPPING

Reference_Line_To Hob Tooth Root _ in

Radius in Hob Tooth Root _ in

Ref_Line to Hob SAP 0.0517 in

Normal_Space Width at Hob SAP 0.1195 in

PINION SHAPER

Number_of Teeth _

Outside_Diameter _ in

Normal_Tooth_Thickness _ in

Tip_Radius - Normal Plane _ in

Protuberance - Normal Plane _ in

Center distance with shaper cutter NA in

Start_of_active_profile_diameter in


12


GEAR SHAPER

Number_of Teeth 40

Outside_Diameter 4.2500 in

Normal_Tooth_Thickness 0.1571 in

Tip_Radius - Normal Plane 0.0250 in

Protuberance - Normal Plane 0.0000 in

Center distance with shaper cutter 1.3038 in

Start_of_active_profile_diameter 3.8250 in

The model would have notified us if the normal tooth thickness at the effective outside diameter (in this case the actual OD) was less than 0.025 inch or 0.25/NDP. (For fine pitch gears we would be notified if the normal tooth thickness was less then 0.275/NDP.) If the gear is case hardened the tooth tip may be over hardened and brittle. (If the angle between the tangent line to the outside diameter and the normal to the involute at this point is greater than 90 degrees the effect of load combined with the small tooth thickness is sometimes ignored. The model will override the warning if this is the case.) Contact will start on the pinion at 4.559 degrees roll angle (output variable 85), since there is no contact below the finished involute profiles. (This is pretty close to the base circle and may not be a satisfactory design.) We will plot 3 teeth on each gear with tooth #1 on the pinion at the start of active profile. In the Plot Configuration tab, click the radio button for “Actual SAP 4.559 deg”. Then enter 3 teeth to be plotted and tooth 1 as the driver tooth number. Toggle to TK Solver and pick the “mesh” plot from the toolbar Plots list. Your plot should look like Figure 1B. The small “tic” marks indicate the intersection of the involute and fillet.


13

Fig. 1B

Note that contact is just taking place at the root of the driver (pinion) and the tip of the driven (internal gear). One base pitch away down the line of action a second tooth is also in contact. We wish to check the pinion tooth at the first point of contact for undercut and interference. To do this, go to the plot subsheet for “mesh”. (Put the pointer anywhere on the plot and click the right mouse button.) Enter yes for “Display Scale:” and “Display Grid:”. The plot subsheet is shown in Figure 1C.


14

Fig. 1C

The plot should then look like Figure 1D.


15

Fig. 1D

From the plot grid we see that the area we are interested in is bounded by -.2, 0 on the X-Axis and 0,.1 on the Y-Axis. Enter these values on the subsheet, as shown in Figure 1E.


16

Fig. 1E

From the plot, Figure 1F, there seems to be no undercut on the pinion and the gear tooth tip is contacting on the involute profile.


17

Fig. 1F

To be sure we will “zoom” in some more. This time we will use an “X” range of -.13 to -.11 and a “Y” range of .03 to .04.


18

Fig. 1G

Figure 1G shows that the tooth tip of the gear is indeed in contact with the involute flank of the pinion and there is no undercut at the pinion start of active profile. The start of active profile, however, is very close to the involute/fillet junction and a small dimensional change would put the contact below the junction. Now we will take a quick look at the start of active profile on the gear although we expect no difficulty here. The lowest point of single tooth contact on the pinion for pinion tooth #1 will place pinion tooth #2 in contact at the tooth tip. This is also the start of active profile on the gear. In the Plot Configuration tab of the data input form, enter 2 teeth to be plotted, 20.260 degrees roll for the pinion and 1 for pinion tooth number default to 1. Toggle to TK Solver and the subsheet. Set “Display Scale” to “No” and blank the X and Y axis min-max on the plot subsheet. The plot is shown in Figure 1H.

NOTE: The same thing could be accomplished by setting the pinion roll angle to 50.2601 degrees (outside diameter) and the pinion tooth number to 2.

The plot, Figure 1H, indicates that contact at the gear SAP (last point of contact) is well down on the flank below the involute/fillet intersection and we need look no further.


19

Fig. 1H

As a matter of interest we will confirm that at the operating pitch point, 17.276 degrees roll, we have two teeth in contact with this low center distance design. (The zone of single tooth contact, of course, extends from 20.260 to 34.558 degrees.) Figure 1I shows the teeth in contact at the operating pitch point. Fig. 1I


20

Example 2 This is a high contact ratio (HCR) spur gear set with a profile contact ratio greater than 2. The pinion is hobbed and the gear is shaped. An HCR set has three teeth in contact in a zone just after the start of action, in a zone around the operating pitch diameter and in a zone near the end of contact. There are two teeth in contact in the zones separating the three tooth zones. HCR spur gears have the advantage of quieter operation and somewhat higher capacity than “standard” spurs with a contact ratio between 1 and 2. However, it is essential that load sharing take place between the teeth. This requires high accuracy and, in some cases, properly designed profile modifications. If for any reason the total load is applied to the tip of a tooth, tooth scoring, pitting or breakage likely will occur. The specific sliding ratio tends to be higher because of the relatively large distance from the operating pitch point to the start and end of action. We will assume that the inside diameter of the gear has not yet been designated. We will solve the model first with the calculated default value to obtain the minimum possible inside diameter to avoid involute interference with the pinion. Enter the input data as shown in Figure 2A. The inputs and outputs for the solved model are shown in Report 2A.


21

Fig. 2A

Report 2A





22




NORMAL PLANE





NORMAL PLANE PINION







TRANSVERSE PLANE









23


COMMON



Axial pitch _ in




PINION



















24




Lead _ in



INTERNAL GEAR





















25



Lead _ in



OPERATING DATA

Change in Operating CD from "Std" CD 0.0000 in




















26










PLOT CONFIGURATION






PINION ROLL ANGLES



Lowest_single tooth contact-Spur deg

Lowest_double tooth contact-HCRS 6.445 deg


Highest_single tooth contact-Spur deg

Highest_double tooth contact-HCRS 29.296 deg




27


GEAR ROLL ANGLES




CONTACT DATA












PINION HOB

Hob type n








28







PINION HOB TOPPING





GEAR SHAPER

Number_of Teeth 29






Start_of_active_profile_diameter 2.7763 in The minimum ID to avoid involute interference is 5.7930 inches. We will add about 0.2/Pitch to obtain ample clearance between the interference diameter and the inside diameter. (This will tend to keep the contact away from the base circle of the pinion.) Enter 5.813 for the inside diameter and solve. The data input form is shown in Figure 2B, the solved model in Report 2B.


29

Fig. 2B


30

Report 2B






NORMAL PLANE





NORMAL PLANE PINION







TRANSVERSE PLANE



31








COMMON



Axial pitch _ in




PINION













32










Lead _ in



INTERNAL GEAR















33









Lead _ in



OPERATING DATA
















34















PLOT CONFIGURATION






PINION ROLL ANGLES




Lowest_double tooth contact-HCRS 6.445 deg


35




Highest_double tooth contact-HCRS 26.418 deg



GEAR ROLL ANGLES




CONTACT DATA












PINION HOB

Hob type n



36












PINION HOB TOPPING





GEAR SHAPER

Number_of Teeth 29








37

The calculated (and actual) profile contact ratio is over 2 and there is no contact below the finished involute for either gear. The approach action, however, is greater than 55% and the maximum specific sliding ratio on the pinion is over 3. (Messages in the model warn you of this.) This is difficult to avoid on HCR gears but increasing the pinion OD and/or the gear ID would help to reduce the approach action and pinion specific sliding ratio. We will not attempt to change this design, as it is only an example. This time we will look at the teeth on each individual gear to check for undercut. Toggle to TK Solver and select the plot “pinion” from the toolbar Plots list. Your plot of the pinion tooth should look like Figure 2B1.

Fig. 2B1


38

Next select “gear”. Your plot of the gear tooth should look like Figure 2B2. There does not appear to be any great problem with the form of the teeth using these cutting tools and tooth proportions at this point, although the SAP on the pinion is very close to the base circle. This is a cause for a pinion specific sliding ratio over 3. (The ID should probably be increased to move away from the base circle while still keeping the contact ratio safely above 2.) Note that “Contact past finished involute?” is “'No”. Next we will look at three teeth in mesh to illustrate that we do indeed have two or three teeth in contact. In the Plot Configuration tab of the data input form, click the radio button for “Actual SAP 2.418 deg”; enter 3 for the number of teeth and 1 for the starting tooth number. Your plot of the mesh should look like Figure 2C.

Fig. 2B2


39

Fig. 2C

Note that all three teeth are in contact along the line of action. Next we will roll the gears to about 1 degree past the lowest double tooth contact point. On the Plot Configuration tab, enter 7.5 for the roll angle. Now your plot of the mesh should look like Fig. 2D. Fig. 2D


40

Only the first two teeth are in contact along the line of action. The third tooth has rolled out of mesh (about 1 degree). For this example we did not check the mesh at the first and last points of contact. This should be done by “zooming” in on these areas, as was done in Example 1.


41

Example 3 This is a 12/66 tooth, 10 normal pitch internal spur gear set proportioned with the usual set of “cookbook” formulas for center distance, outside diameters and tooth thickness. The formulas used are:

Center distance = (Gear Teeth - Pinion Teeth)/(2 Normal Pitch) = 2.7 inches Outside diameter = (Pinion Teeth + 2)/(Normal Pitch) = 1.4 inches Inside diameter = (Gear Teeth - 2)/(Normal Pitch) = 6.4 inches Tooth thickness = PI/(2 Normal Pitch) - 1/2 Total Backlash = 0.1556 inch (for both gears)

The pinion is hobbed with a non-topping “standard” finish hob. The gear is shaped with a “standard” finish shaper cutter. (These “off the shelf” tools are difficult to work with, as they are usually designed for a root clearance of only 0.157/Pitch.) The pinion is the driver. We will check for undercut and interference at the first and last points of contact. Since the pinion is the driver the first point of contact will be at the pinion root (gear tip) and the last point of contact will be at the pinion tip (gear root). The completed data input form is shown in Figure 3A. After solving you should have Report 3A.


42

Fig. 3A


43

Report 3A






NORMAL PLANE





NORMAL PLANE PINION







TRANSVERSE PLANE



44








COMMON



Axial pitch _ in




PINION













45










Lead _ in



INTERNAL GEAR















46









Lead _ in



OPERATING DATA
















47















PLOT CONFIGURATION






PINION ROLL ANGLES






48







GEAR ROLL ANGLES




CONTACT DATA








Contact past finished involute? Yes




PINION HOB

Hob type n



49












PINION HOB TOPPING





GEAR SHAPER

Number_of Teeth 40








50

The model gives us a message warning that there is internal tooth tip contact below the pinion base diameter and that it will cause involute interference unless the pinion is undercut. This indicates that the entered inside diameter is too small. The smallest diameter that will avoid interference is 6.4712 inches. Change the inside diameter to 6.5000 inches and solve again as shown in Report 3B. Report 3B






NORMAL PLANE





NORMAL PLANE PINION







51



TRANSVERSE PLANE








COMMON



Axial pitch _ in




PINION









52














Lead _ in



INTERNAL GEAR











53













Lead _ in



OPERATING DATA











54




















PLOT CONFIGURATION






PINION ROLL ANGLES


55











GEAR ROLL ANGLES




CONTACT DATA












56



PINION HOB

Hob type n












PINION HOB TOPPING





PINION SHAPER

Number_of Teeth _

Outside_Diameter _ in

Normal_Tooth_Thickness _ in

Tip_Radius - Normal Plane _ in

Protuberance - Normal Plane _ in


57

Model Title : Program 60-450 Unit System: US Center distance with shaper cutter NA in

Start_of_active_profile_diameter in

GEAR SHAPER

Number_of Teeth 40







The report tells us that there is no contact below the finished involute on the pinion and no contact above the finished involute on the gear.

We will start at the pinion SAP and plot 2 teeth. Use the Plot Configuration tab in the data input form to set up the plot. See Figure 3B.

Fig. 3B


58

The plot indicates that the contact on the pinion is very close to the involute/fillet intersection. We will “zoom” in and take a closer look. Open the plot subsheet and set the X-axis minimum and maximum to -.2, 0. Set the scale and grid to “Yes.” (See the detailed explanation of these subsheet operations in the discussion of Figures 1C through 1G, above.) The plot should look like Figure 3C. Fig. 3C

Figure 3C indicates that the inside diameter contacts the pinion very close to the involute/fillet junction and should be changed. The pinion has been undercut by the hob. “Cookbook” formulas usually produce a gear set that will turn without binding or jamming, but they do not produce an “optimum” design and will not always produce a usable gear set.


59

Example 4 This is an 8 pitch, 22.5 degree pressure angle, 23 degree helix angle helical gear set. The pinion is cut with a semi-topping hob. The gear is shaped. Both gears have been post processed after cutting and more stock removed from the tooth flanks. Protuberance was used on the hob and the shaper to provide undercut for “runout” of the finishing tool. It is assumed that the tools were properly matched and no steps were cut in the involute flanks or fillet areas. (See UTS Program 500.) Enter the input data as shown in Figure 4A. The solved model is shown in Report 4A. Fig. 4A


60

Report 4A






NORMAL PLANE





NORMAL PLANE PINION







TRANSVERSE PLANE



61








COMMON



Axial pitch 1.0050 in




PINION



Start_Tip Modification 5.3673 in

Roll_at_Start of Tip Modification 31.432 deg

Normal_OD tip relief 0.01199 in

Normal_circular OD tip relief 0.02203 in

Transverse_circular_OD tip relief 0.02411 in






62










Lead 38.1914 in



INTERNAL GEAR












Pointed_tooth diameter (No tip mod) 11.4176 in



63









Lead 88.4432 in



OPERATING DATA
















64















PLOT CONFIGURATION






PINION ROLL ANGLES






65







GEAR ROLL ANGLES




CONTACT DATA











Unmodified Profile Contact Ratio 1.373

PINION HOB

Hob type s



66






Protuberance_angle from flank 10.0000 deg

Protuberance_pressure angle 12.5000 deg

Tip_to_flank/prot intersection 0.0460 in


Reference_Line_to Start Mod Ramp 0.1000 in

Pressure Angle of Mod Ramp 55.5000 deg

PINION HOB TOPPING





GEAR SHAPER

Number_of Teeth 29








67

There is no contact past the finished involute on either gear. However, there will be some undercut due to the protuberance on the hobs. The approach action is a little higher than 55%. It is usually desirable to have more of the contact in the zone of recess (past the operating pitch point) than in the approach zone. We will plot two teeth with tooth #1 at the operating pitch point. Use the Plot Configuration tab of the data input form. Fig. 4B

The plot scale includes the full length of the line of action for both gears. To get a larger scale plot of the teeth we will “zoom” in a little. You may turn on the scale (and the grid) if you wish. An X-axis range from -0.5 to 0.5 should include the teeth. (See instructions for these subsheet operations in the discussion of Figures 1C through 1G, above.) The plot should look like Figure 4C.


68

Fig. 4C

It is left to you to check this gear set for undercut below the contact zone on the pinion, above the contact zone for the gear, and for interference if you wish.


69

Example 5 This is a formed or molded 25/45 tooth, 10 pitch, 20 degree pressure angle, high addendum pinion spur gear internal set. The gears will have a tip radius at the tooth tips of 0.015 inch. In addition, approximately 0.0015 inch tip relief is required at the effective tooth tips of both gears. (Try about 0.002 inch at the normal OD & ID and then adjust to get about 0.0015 inch at the effective OD & ID.) Enter the input data as shown in Figure 5A, overriding defaults when necessary. Answer no to entering tip chamfer data, yes to other data entries. The solved model is shown in Report 5A. Fig. 5A


70

Report 5A



Hobbed ('hob), Shaped ('shp), Formed frm ('frm)


Shaped ('shp) or Formed ('frm) frm

NORMAL PLANE





NORMAL PLANE PINION







TRANSVERSE PLANE



71








COMMON



Axial pitch _ in




PINION



Start_Tip Modification 2.7250 in


Normal_OD tip relief 0.00200 in

Normal_circular OD tip relief 0.00238 in

Transverse_circular_OD tip relief 0.00238 in


Normal_effective OD tip relief 0.00153 in




72










Lead _ in



INTERNAL GEAR




Start_Tip_Modification 4.3750 in


Normal_ID tip relief 0.00250 in

Normal_circular_ID tip relief 0.00254 in

Transverse_circular_ID tip relief 0.00254 in


Normal_effective_ID tip relief 0.00151 in





73









Lead _ in



OPERATING DATA

Change in Operating CD from "Std" CD -0.0500 in















74















PLOT CONFIGURATION




Pinion_contact roll angle of 25.783 deg


PINION ROLL ANGLES






75







GEAR ROLL ANGLES




CONTACT DATA


Arc_of_approach -6.398 deg









Unmodified Profile Contact Ratio 0.911

FORMED PINION




76




Radial_tip chamfer (w/o mod) 0.0000 in

Normal_tip radius 0.0150 in

Normal_tip_relief exponent 1.5000

FORMED GEAR

Normal_fillet radius 0.0508 in

Radial_tip chamfer (w/o mod) 0.0000 in

Normal_tip radius 0.0150 in

Normal_tip_relief exponent 1.5000 Note: It took a couple of tries to get the tip relief to come out at about 0.0015 inch at the effective OD and ID. The amount of normal tip relief at the pinion effective outside diameter (where the tip radius starts) is 0.00153 inch. At the gear effective inside diameter the normal tip relief is 0.00151 inch. These values result in involute/fillet intersection diameters that are below the start of active profile and above the base diameter for the pinion and above the start of active profile on the gear. The root clearance for the pinion is 0.0329 inch and for the gear 0.0291 inch. If desired, further adjustment can easily be made. However, it may be helpful to check a plot of the gears before reducing the root clearance to small values in an attempt to reduce the root bending stress. With full root radius gears and small root clearance the possibility of fillet interference with the mating gear tip must be checked. We will plot 3 teeth for each gear with the first pinion tooth at the start of contact (output variable 78 of Report 5A). Use the Plot Configuration tab of the data input form. Figure 5B is the plot of the teeth. (Plots should be made at various points through the mesh to insure that no fillet interference exists.)


77

Fig. 5B


78

Example 6 This is a 16/64 spur gear set, with a module of 1 mm, molded from thermoplastic materials. The housing and the 64 tooth gear are molded from acetal and the 16 tooth pinion is molded from nylon. The range of operating temperatures ranges from -10 degrees C to +60 degrees C. The relative humidity will range from dry to 100% for long periods. The design must be capable of operation at maximum effective center distance without running out of backlash and without root-tip interference. The design must also retain a contact ratio over one at the minimum effective center distance.

The first step is to find the minimum and maximum effective center distances under the ambient conditions. For design purposes the gear dimensions are subjected to tolerance variations at an assembly temperature of 20 degrees C but not to dimensional changes due to temperature changes and moisture absorption. All such changes are accounted for in the “effective” center distance. The minimum and maximum “effective” center distances will not actually occur but are used to check the gear set for interference and contact ratio under conditions which would have the same effect as the actual operating conditions.

The “standard” center distance for this gear set is 24 mm. Since this is a 4 to 1 ratio we will design the 16 tooth pinion on a 17 tooth blank to avoid undercut, increase pinion strength, and keep the contact well above the base diameter. (Contact too near the base diameter will result in high specific sliding ratios causing a large temperature rise in the mesh.) Our “design” nominal center distance will then be 23.5 mm.

For the above ambient conditions the minimum effective center distance is found to occur at hot and dry conditions. The minimum effective center distance is 23.393 mm. The maximum effective center distance, under cold and humid conditions, is 23.728 mm. (See models 60-108, 60-100 and 60-146.)

Next we will design the gear set at maximum effective center distance and minimum tooth tip radius along with maximum tooth thickness. We will use maximum outside diameter for the pinion and minimum inside diameter for the gear. The backlash will be set to 0.050 mm as this is the tightest mesh condition. (If the backlash is too small, the deflection of the teeth will cause contact on the non-driving side of the teeth and jamming and excessive wear will result.) These conditions will produce the greatest chance of interference and the minimum root clearance. Since we will lose contact ratio when we go to minimum effective center distance we will use the nominal outside diameter of the pinion and the nominal inside diameter of the gear for the design. We will use a negative tolerance to obtain the minimum OD of the pinion and a positive tolerance to obtain the maximum ID of the gear.


79

Using full depth proportions will then give us an OD of 19 mm for the pinion and an ID of 62 mm for the gear. This represents an addendum change of +0.5 mm for the pinion. The tooth thickness of the pinion would then be PI*module/2+2*(addendum change)*tangent(normal pressure angle). (This calculation can be done directly in the TK Solver Variable Sheet input column by entering pi()*1/2+2*.5*tand(20).) The pinion tooth thickness will then be 1.935 mm. The model will calculate the tooth thickness of the gear since we will enter a backlash of 0.050 mm. The tooth tip radius will be 0.17 mm to 0.21 mm. At maximum effective center distance we need to use a tip radius of 0.17 mm to get the largest effective outside diameter on the pinion and the smallest effective inside diameter on the gear. Enter the data as shown in Figure 6A; override the defaults if necessary. (Be sure the model is in metric units.) After entering the data and solving, you should have the data in Report 6A.


80

Fig. 6A

Report 6A

Model Title : Program 60-450 Unit System: Metric




81




NORMAL PLANE




Base pitch 2.9521 mm

NORMAL PLANE PINION

Finished tooth thickness at Ref PD 1.9350 mm

Total circular finish stock on tooth 0.0000 mm thickness



Finished_space width at Ref PD 1.7966 mm


TRANSVERSE PLANE




Base_pitch 2.9521 mm

Pinion_tooth thickness at Ref PD 1.9350 mm

Gear_tooth thickness at Ref PD 1.3450 mm

Gear_space width at Ref PD 1.7966 mm


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COMMON



Axial pitch _ mm

Operating_center distance 23.72800 mm

Standard_center distance 24.0000 mm

Net face width 20.0000 mm

PINION

Outside Diameter 19.0000 mm

Normal_top land width 0.2285 mm

Start_Tip Modification NA mm


Normal_OD tip relief NA mm

Normal_circular OD tip relief NA mm

Transverse_circular_OD tip relief NA mm

Effective_outside diameter 18.8630 mm

Normal_effective OD tip relief NA mm

Normal_tooth_thickness_at_Eff OD 0.5013 mm

Pointed tooth diameter (No tip mod) 19.5065 mm

Reference PD 16.0000 mm



Normal_TT_at_inv/fill intersection dia 2.0483 mm

Minimum_fillet radius 0.4490 mm

Root diameter 14.3412 mm


83


Whole depth of tooth (from Eff OD) 2.3294 mm

Base_diameter 15.0351 mm

Lead _ mm

Area of space (Normal section) 3.94 mm^2

Area of tooth (Normal section) 3.69 mm^2

INTERNAL GEAR

Inside (minor) Diameter 62.0000 mm


Minimum ID w/o involute interference 61.9242 mm

Start_Tip_Modification NA mm


Normal_ID tip relief NA mm

Normal_circular_ID tip relief NA mm

Transverse_circular_ID tip relief NA mm

Effective_inside diameter 62.2514 mm

Normal_effective_ID tip relief NA mm

Normal_Tooth Thickness at eff ID 0.7605 mm

Pointed_tooth diameter (No tip mod) Below_BD mm


Inv/fillet intersection dia (TIF) 65.7670 mm




Major diameter 66.5003 mm

Whole depth of tooth (from Eff ID) 2.2502 mm


84



Lead _ mm



OPERATING DATA

Change in Operating CD from "Std" CD -0.2720 mm

Working depth of active flanks 2.0338 mm

Total_working depth 2.2280 mm

Transverse_circular backlash 0.0502 mm







Pitch_diameter 15.8187 mm

Normal_tooth thickness 1.9754 mm

Transverse_tooth thickness 1.9754 mm


Start of active profile (SAP) 15.0932 mm

Root_Clearance 0.1014 mm





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Normal_space width 2.0256 mm


Transverse_space width 2.0256 mm


Start_of active profile (SAP) 65.5764 mm

Root_clearance 0.0222 mm


PLOT CONFIGURATION






PINION ROLL ANGLES











86


GEAR ROLL ANGLES




CONTACT DATA








Contact past finished involute? Yes




FORMED PINION


Tip to Reference Line 1.3300 mm

Tooth thickness at Reference Line 1.5710 mm

Tip_radius 0.4300 mm

Radial_tip chamfer (w/o mod) 0.0000 mm

Normal_tip radius 0.1700 mm

Normal_tip_relief exponent _


87


FORMED GEAR

Normal_fillet radius 0.6080 mm



Normal_tip_relief exponent _ The lowest point of contact on the pinon (SAP) is 5.04 degrees roll. The roll angle at the involute-fillet intersection point is at about 8.68 degrees and so we have contact below the finished involute on the pinion. In many cases contact may occur down to the involute-fillet intersection point when at the maximum effective center distance. The effective outside or inside diameter of the mate will then be in the root fillet area below the involute portion of the tooth. Interference must be checked by plotting the tooth forms in the proper relation to each other and checking for contact between the tooth tips and fillets. Since internal gears tend to contact external gears quite deeply, we will add 0.3 mm to the ID of the internal (to get about 0.25 mm pinion root clearance). Adjust the “Inv/fillet intersection dia” for the internal gear to obtain a gear root clearance of about 0.25 mm after adding 0.3 mm to the OD of the pinion. Set the normal fillet radius at .332. The complete data input form is shown in Figure 6B, and the solved model in Report 6B.


88

Fig. 6B

Report 6B





89




NORMAL PLANE





NORMAL PLANE PINION







TRANSVERSE PLANE









90


COMMON



Axial pitch _ mm




PINION



















91




Lead _ mm



INTERNAL GEAR





















92



Lead _ mm



OPERATING DATA





















93










PLOT CONFIGURATION






PINION ROLL ANGLES











94


GEAR ROLL ANGLES




CONTACT DATA












FORMED PINION








FORMED GEAR






95

The lowest point of contact on the pinion is now above the involute-fillet intersection point. We will plot 2 teeth for each gear with the first pinion tooth at the start of contact. Use the Plot Configuration tab of the data input form. The plot is shown in Figure 6C. Fig. 6C

We don't need a plot of the entire line of action, so we will cut down the plot range to get a better picture. In the “mesh” plot subsheet, set the X-axis minimum-maximum at -.2, .2. (See Example 1 to review how to do this.) Figure 6D shows the plot.


96

Fig. 6D

We need to “zoom” in on the root of the pinion for a closer look to make sure there is no interference. Set the X-axis minimum-maximum at -.06,-.02 and the Y-axis minimum-maximum at .01, .04. You might also set the scale and grid to “Yes.” The plot is shown in Figure 6E. It shows that there is no interference at the first point of contact.


97

Fig. 6E

Now we will roll the gear set to the last point of contact at the tip of the pinion tooth. Fig. 6F


98

The gear set must also be checked at min effective center distance and max tip radius along with min pinion OD, max gear ID and min tooth thickness. This condition will produce the lowest contact ratio and the maximum backlash. Figure 6G shows the completed input data form for this analysis and Report 6G shows the solved model. Fig. 6G


99

Report 6G






NORMAL PLANE





NORMAL PLANE PINION






Total circular finish stock on tooth 0.0000 in Thickness

TRANSVERSE PLANE



100








COMMON



Axial pitch _ mm




PINION













101










Lead _ mm



INTERNAL GEAR















102









Lead _ mm



OPERATING DATA
















103















PLOT CONFIGURATION






PINION ROLL ANGLES






104







GEAR ROLL ANGLES




CONTACT DATA


Arc_of_approach -4.940 deg










FORMED PINION




105







FORMED GEAR




Normal_tip_relief exponent _ We will plot 2 teeth at the pinion first point of contact. Use the Plot Configuration tab of the data input form. The contact ratio is 1.045 at this extreme condition and so we still have conjugate action. Although stresses may be quite high, the backlash has increased to 0.299 mm. A range on the X axis of -.1 to .2 should cover the plot.


106

Fig. 6H

Next, 2 teeth with contact at the last point of contact on tooth #2.

Fig. 6I

This gear set would function correctly in that conjugate action would occur at both extreme conditions. The gear set would also have to meet load and life requirements along with noise requirements.

program 60-450—internal gear line of action tooth plot ... · pdf fileinvolute spur and...

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